J. Shimada, H. Kaneko, and T. Takada. Performance of fast multipole methods for calculating electrostatic interactions in biomacromolecular simulations. J. Comp. Chem., 15(1):28, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....thereby avoiding a reverse FFT. The forces require only a local summation over the grid since the density of the Gaussian screening function is highly localized. In [100] the Poisson equation is solved by a multi grid approach (O(N) The particle particle particle mesh multi pole expansion [104] (P M PME, O(N log N) is basically an extension of the P M [54] method for periodic boundary conditions using multi pole expansions. The cell multi pole method [28] CMM, O(N) shares the key features of the multi pole methods, but it uses Cartesian coordinates only, unlike the other ....

J. Shimada, H. Kaneko, and T. Takada. Performance of fast multipole methods for calculating electrostatic interactions in biomacromolecular simulations. J. Comp. Chem., 15(1):28, 1994.

....for efficient three dimensional capacitance calculations, while Pringle [50] does three dimensional vortex simulations on a parallel machine. Ding, Karasawa and Goddard [17, 16] use the three dimensional Fast Multipole Method for the simulation of periodic materials. Shimada, Kaneko and Takada [55, 56] present a combination of the Fast Multipole Method with the Hockney and Eastwood method. 1.2 Multi Level Time Stepping The time stepping method most often used in molecular dynamics is the Verlet scheme, which is to mathematicians known as Stormer s method. Especially popular is the velocity ....

....summations we used an Ewald summation approach, as described in [48, 49, 14] with the general theory given in [13, 34] Schmidt and Lee [54] implemented the Fast Multipole Method with periodic boundary conditions. They did not give a derivation of their formula, and Shimeda, Kaneko, and Takada [56] pointed out typographical errors in their equations. In the following we give a derivation of the formulas used in our implementation. Definition 3.3.1 The Schwartz space S of rapidly decreasing functions is composed of all complexvalued functions (x) 2 C 1 (R n ) with sup x2R n Gamma ....

[Article contains additional citation context not shown here]

Jiro Shimada, Hiroki Kaneko, and Toshikazu Takada. Performance of fast multipole methods for calculating electrostatic interactions in biomacromolecular simulations. Journal of Computational Chemistry, 15(1):28--43, 1994.

....prescribed error requirement. 3 Table 2 Some previous implementations of O(N ) N body methods. Author Method Degree of separation Use of supernodes Hierarchy depth Greengard Rokhlin [5] GR 2 No blog N 8 c Zhao [14] Zhao 2 Yes blog N 8 c Schmidt Lee [11] GR 1 No 3,4,5 Shimada et al. [12] GR 1,2 No log N 8 Leathrum [10] GR 1,2,mixed 1 and 2 Yes and No blog N 8 c Esselink [3] GR 2 Yes 2,3,4 The scheme is developed and verified for uniform distributions of particles. The impact on the error of non uniform particle distributions is studied through some model distributions. Our ....

J. Shimada, H. Kaneko, and T. Takada. Performance of fast multipole methods for calculating electrostatic interactions in biomacromolecular simulations. J. Comp. Chem., 15(1), 1994.

....of particles. Schmidt and Lee [16] examined the accuracy and the execution time for a few combinations of hierarchy depth and number of terms in the multipole expansions for a 3 D GR method with one separation (i.e. the near field consists of only nearest neighbor elements) Shimada et al. [17] examined a few combinations of hierarchy depth, one separation and two separation (i.e. the near field consists of nearest neighbor elements and their nearest neighbors as well) Using a shared memory implementation of the GR method, Leathrum [13] numerically examined the tradeoffs between ....

....1.1 Characteristics of some previous implementations of multipole like O(N) N body methods. Author Method Degree of Use of Hierarchy separation supernodes depth Greengard Rokhlin [7, 9] GR 2 No blog N 8 c Zhao [21] Zhao 2 Yes blog N 8 c Schmidt Lee [16] GR 1 No 3,4,5 Shimada et al. [17] GR 1,2 No log N 8 Leathrum [13] GR 1,2,mixed 1 and 2 Yes and No blog N 8 c Esselink [5] GR 2 Yes 2,3,4 into two parts: OE total = OE near Gammafield OE far Gammaf ield ; 2.1) where OE near Gammafield is the potential due to nearby particles and OE far Gammaf ield is the potential due to ....

J. Shimada, H. Kaneko, and T. Takada. Performance of fast multipole methods for calculating electrostatic interactions in biomacromolecular simulations. Journal of Computational Chemistry, 15(1):28--43, 1994.

....of particles. Schmidt and Lee [SL91] examined the accuracy and the execution time for a few combinations of hierarchy depth and number of terms in the multipole expansions for a 3 D GR method with one separation (i.e. the near field consists of only nearest neighbor elements) Shimada et al. SKT94] examined a few combinations of hierarchy depth, one separation and two separation (i.e. the near field consists of nearest neighbor elements and their nearest neighbors as well) Using a shared memory implementation of the GR method, Leathrum [Lea92] numerically examined the tradeoffs ....

....other approximations for a wide range Chapter 7. Accuracy of Anderson s Method 111 Author Method Degree of Use of Hierarchy separation supernodes depth Greengard Rokhlin [GR87b, GR88] GR 2 No blog N 8 c Zhao [Zha87] Zhao 2 Yes blog N 8 c Schmidt Lee [SL91] GR 1 No 3,4,5 Shimada et al. SKT94] GR 1,2 No log N 8 Leathrum [Lea92] GR 1,2,mixed 1 and 2 Yes and No blog N 8 c Esselink [Ess94] GR 2 Yes 2,3,4 Table 7.1: Characteristics of some previous implementations of multipole like O(N) N body methods. of errors. Esselink [Ess94] studied the arithmetic complexity, accuracy and ....

Jiro Shimada, Hiroki Kaneko, and Toshikezu Takada. Performance of fast multipole methods for calculating electrostatic interactions in biomacromolecular simulations. Journal of Computational Chemistry, 15(1):28--43, 1994.

....capacitance calculations, while Pringle [Pri94] does three dimensional vortex simulations on a parallel machine. Ding, Karasawa and Goddard [DKu92a, DKu92b] use the three dimensional Fast Multipole Method for the simulation of periodic materials. Shimada, Kaneko and Takada [SKT93, SKT94] present a combination of the Fast Multipole Method with the Hockney and Eastwood method. In this report we focus on the implementation of the Fast Multipole Method in three dimensions with periodic boundary conditions. To achieve an efficient implementation we reformulate Greengard s formulae ....

....to an Ewald summation approach, as described in [NdW57, NdW58, dWN58] with the general theory given in [dLPS80, Hey81] Schmidt and Lee [SL91] implemented the Fast Multipole Method with periodic boundary conditions. They did not give a derivation of their formula, and Shimeda, Kaneko, and Takada [SKT94] pointed out typographical errors in their equations. In the following we give a derivation of the formulae used in our implementation. Definition 4.1 The Schwarz space S of rapidly decreasing functions is composed of all complex valued functions (x) 2 C 1 (R n ) with sup x2R n i 1 ....

[Article contains additional citation context not shown here]

Jiro Shimada, Hiroki Kaneko, and Toshikazu Takada. Performance of fast multipole methods for calculating electrostatic interactions in biomacromolecular simulations. Journal of Computational Chemistry, 15(1):28--43, 1994.

.... the Fast Multipole Method started to appear, nearly all of which were concerned with the application of the Fast Multipole Method to different problems or the implementation on parallel machines [Kat89, Ess94, LB92, BHE 94, ZJ91, SL91, WS92, WS93, Nab93, NKLW94, Pri94, DKu92a, DKu92b, SKT93, SKT94] In this article we focus on the implementation of the Fast Multipole Method in three dimensions with periodic boundary conditions. To achieve an efficient implementation we reformulate Greengard s formulae in section 1. In section 2 we derive an Ewald type formula for the contributions of the ....

....to an Ewald summation approach, as described in [NdW57, NdW58, dWN58] with the general theory given in [dLPS80, Hey81] Schmidt and Lee [SL91] implemented the Fast Multipole Method with periodic boundary conditions. They did not give a derivation of their formula, and Shimeda, Kaneko, and Takada [SKT94] pointed out typographical errors in their equations. In the following we give a derivation of the formulae used in our implementation. Definition 2.1 The Schwartz space S of rapidly decreasing functions is composed of all complex valued functions (x) 2 C 1 (R n ) with sup x2R n i 1 ....

Jiro Shimada, Hiroki Kaneko, and Toshikazu Takada. Performance of fast multipole methods for calculating electrostatic interactions in biomacromolecular simulations. Journal of Computational Chemistry, 15(1):28--43, 1994.

....Our new algorithm will draw on techniques from the FMA, but applied novelly to create an online 2 nonparametric estimation algorithm. A brute force online algorithm contains a double 1 The fast multipole algorithm and modifications thereof for a wide variety of physics problems can be found in [2, 3, 5, 6, 8, 9, 12, 16, 17, 21, 23, 25]. 2 An online algorithm is defined as an algorithm operating in a simulated real time environment, where the input dataset grows with each new point seen (or grows in blocks) and recomputes the estimates by adding input points sequentially. z1 z2 z3 zn z Figure 1: Multipole expansion. A ....

J. Shimada, H. Kaneko, and T. Takada. Performance of fast multipole methods for calculating electrostatic interactions in biomacromolecular simulations. Journal of Computational Chemistry, 15(1):28--43, 1994.

....of the O(n 2 ) terms in the last sum in the potential of Table 1 need to be calculated. To make full use of the resulting sparsity (which varies from iteration to iteration) efficient data structures must be maintained; see, e.g. Schreiber et al. 281] More recently, fast multipole expansions [25, 120, 284] and variants [70, 91] of the Ewald method (Ewald [92] were used as an alternative to cut off methods, with a significant increase in quality York et al. 353, 354] An improvement of a divide and conquer method by Appel [8] for fast potential evaluation is discussed in Xue et al. 352] While ....

....potentials. One of the obvious difficulties is that because of the high dimension and the expensive evaluation of the potential, even local optimization is slow. For large molecules, this is the case even when the potential calculations are speeded up using fast multipole expansions [25, 120, 284] or potential cutoffs. The fastest optimization methods, the adopted basis Newton Raphson (ABNR) method and truncated Newton (TN) methods, employed, e.g. in CHARMM [33, 76] combine elements of Newton s method with reduced subspace techniques to reduce storage requirements and 26 A. NEUMAIER ....

J. Shimada, H. Kaneko and T. Takada, Performance of fast multipole methods for calculating electrostatic interactions in biomacromolecular simulations, J. Comp. Chem. 15 (1994), pp. 28--43.

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